The Four Colour Theorem

One day I was browsing the internet looking for some captivating mathematics theorems and while doing this I came across the four colour theorem, which didn’t seem to be enthralling at first, but after realizing it’s importance I was intrigued by it and decided to write about it and let everyone know about this important theorem which plays an important role in our daily lives.

The four color theorem states that any map--a division of the plane into any number of regions--can be colored using no more than four colors in such a way that no two adjacent regions share the same color. The four color theorem is particularly notable for being the first major theorem proved by a computer. The conjecture that any map could be coloured using only four colours first appeared in a letter from Augustus De Morgan, professor of mathematics at the new University College London, to his friend William Rowan Hamilton, the famous Irish mathematician. It had been suggested to De Morgan by one of his students, Frederik Guthrie.The problem, so simply described, but so tantalizingly difficult to prove, caught the imagination of many mathematicians at the time. The application of mathematical induction to the problem was led after Arthur Cayley wrote the first paper on the conjecture. Fallacious proofs were given independently by Kempe and Tait. Both proofs were accepted for a decade but were later proven wrong by Heawood. The four colour theorem was then finally obtained by Appel and Haken in 1977, who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. However, no flaws have yet been found, so the proof appears valid and the four colour theorem does exist.

The four colour theorem plays a big part in our lives and one of its most notable applications is in the mobile phone masts, as these masts cover certain areas with some overlap, they can’t all transmit on the same frequency. The simplest solution to the problem was to ensure that no two masts that overlap each other have the same frequency and give them all different frequencies. But, as the government owns all frequencies and charges for each, the firms want to use the minimum possible number of frequencies. The area covered can be drawn as a map and different frequencies can be represented as colours and the four colour theorem can be applied. If this theorem did not exist then mobile phones would have been much more expensive and not everyone would be able to afford them. Another use of this theorem is in scheduling.Consider a University, where you are trying to schedule times for all of the final exams. Some students are taking more than one class, so you want to make sure they don't have two exams scheduled at the same time. However, you want your exam writing period to be as short as possible, to run as many exams concurrently as you can.You can represent this as a Graph coloring problem: you make 𝐺=(𝑉,𝐸) where each class is a vertex, and an edge between vertices any time a two classes contain the same student. Your colors will represent different exam time slots. The minimum number with which you can color that graph is the smallest number of timeslots you need to write all your exams.

I think that the four colour theorem is an essential part of our lives and has made it easier to live in the twenty-first century.